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Cluster algebras Teichmuller Theory Geodesic lengths Quatisation Decorated character variety
Quantum cluster algebras from geometry
Marta Mazzocco
Based on Chekhov-M.M. arXiv:1509.07044 andChekhov-M.M.-Rubtsov arXiv:1511.03851
Marta Mazzocco Quantum cluster algebras from geometry
Cluster algebras Teichmuller Theory Geodesic lengths Quatisation Decorated character variety
Ptolemy Relation
aa′ + bb′ = cc ′
c
a
bc ′
a′
b′
Marta Mazzocco Quantum cluster algebras from geometry
Cluster algebras Teichmuller Theory Geodesic lengths Quatisation Decorated character variety
Ptolemy Relation
•
•
••
•
• x ′1x1
x2
x3
x4
x5
x6
x7
x8
x9
(x1, x2, x3, x4, x5, x6, x7, x8, x9) → (x ′1, x2, x3, x4, x5, x6, x7, x8, x9)
x1x′1 = x9x7 + x8x2
Marta Mazzocco Quantum cluster algebras from geometry
Cluster algebras Teichmuller Theory Geodesic lengths Quatisation Decorated character variety
Ptolemy Relation
•
•
••
•
• x ′1x1
x2
x3
x4
x5
x6
x7
x8
x9
(x1, x2, x3, x4, x5, x6, x7, x8, x9) → (x ′1, x2, x3, x4, x5, x6, x7, x8, x9)
x1x′1 = x9x7 + x8x2
Marta Mazzocco Quantum cluster algebras from geometry
Cluster algebras Teichmuller Theory Geodesic lengths Quatisation Decorated character variety
Ptolemy Relation
•
•
••
•
• x ′1x1
x2
x3
x4
x5
x6
x7
x8
x9
(x1, x2, x3, x4, x5, x6, x7, x8, x9) → (x ′1, x2, x3, x4, x5, x6, x7, x8, x9)
x1x′1 = x9x7 + x8x2
Marta Mazzocco Quantum cluster algebras from geometry
Cluster algebras Teichmuller Theory Geodesic lengths Quatisation Decorated character variety
Ptolemy Relation
•
•
••
•
• x ′1x1
x2
x3
x4
x5
x6
x7
x8
x9
(x1, x2, x3, x4, x5, x6, x7, x8, x9) → (x ′1, x2, x3, x4, x5, x6, x7, x8, x9)
x1x′1 = x9x7 + x8x2
Marta Mazzocco Quantum cluster algebras from geometry
Cluster algebras Teichmuller Theory Geodesic lengths Quatisation Decorated character variety
Ptolemy Relation
•
•
••
•
• x ′1
x2
x3
x4
x5
x6
x7
x8
x9
(x1, x2, x3, x4, x5, x6, x7, x8, x9) → (x ′1, x2, x3, x4, x5, x6, x7, x8, x9)
x1x′1 = x9x7 + x8x2
Marta Mazzocco Quantum cluster algebras from geometry
Cluster algebras Teichmuller Theory Geodesic lengths Quatisation Decorated character variety
Ptolemy Relation
•
•
••
•
• x ′1
x2
x3
x4
x5
x6
x7
x8
x9
(x1, x2, x3, x4, x5, x6, x7, x8, x9) → (x ′1, x2, x3, x4, x5, x6, x7, x8, x9)
(x ′1, x2, x3, x4, x5, x6, x7, x8, x9) → (x ′1, x2, x′3, x4, x5, x6, x7, x8, x9)
Marta Mazzocco Quantum cluster algebras from geometry
Cluster algebras Teichmuller Theory Geodesic lengths Quatisation Decorated character variety
Ptolemy Relation
•
•
••
•
• x ′1
x2 x ′2
x3
x4
x5
x6
x7
x8
x9
(x1, x2, x3, x4, x5, x6, x7, x8, x9) → (x ′1, x2, x3, x4, x5, x6, x7, x8, x9)
(x ′1, x2, x3, x4, x5, x6, x7, x8, x9) → (x ′1, x′2, x3, x4, x5, x6, x7, x8, x9)
Marta Mazzocco Quantum cluster algebras from geometry
Cluster algebras Teichmuller Theory Geodesic lengths Quatisation Decorated character variety
Cluster algebra
We call a set of n numbers (x1, . . . , xn) a cluster of rank n.
A seed consists of a cluster and an exchange matrix B, i.e. askew–symmetrisable matrix with integer entries.
A mutation is a transformationµi : (x1, x2, . . . , xn) → (x ′1, x
′2, . . . , x
′n), µi : B → B ′ where
xix′i =
∏j :bij>0
xbijj +
∏j :bij<0
x−bijj , x ′j = xj ∀j = i .
Definition
A cluster algebra of rank n is a set of all seeds (x1, . . . , xn,B)related to one another by sequences of mutations µ1, . . . , µk . Thecluster variables x1, . . . , xk are called exchangeable, whilexk+1, . . . , xn are called frozen. [Fomin-Zelevnsky 2002].
Marta Mazzocco Quantum cluster algebras from geometry
Cluster algebras Teichmuller Theory Geodesic lengths Quatisation Decorated character variety
Example
Cluster algebra of rank 9 with 3 exchangeable variables x1, x2, x3and 6 frozen ones x4, . . . , x9.
•
•
••
•
• x ′1x1
x2
x3
x4
x5
x6
x7
x8
x9
(x1, x2, x3, x4, x5, x6, x7, x8, x9) → (x ′1, x2, x3, x4, x5, x6, x7, x8, x9)
x1x′1 = x9x7 + x8x2
Marta Mazzocco Quantum cluster algebras from geometry
Cluster algebras Teichmuller Theory Geodesic lengths Quatisation Decorated character variety
Outline
Are all cluster algebras of geometric origin?
Introduce bordered cusps
Geodesics length functions on a Riemann surface withbordered cusps form a cluster algebra.
All Berenstein-Zelevinsky cluster algebras are geometric
Marta Mazzocco Quantum cluster algebras from geometry
Cluster algebras Teichmuller Theory Geodesic lengths Quatisation Decorated character variety
Teichmuller space
For Riemann surfaces with holes:
Hom (π1(Σ),PSL2(R)) /GL2(R).
Idea:
Teichmuller theory for a Riemann surfaces with holes is wellunderstood.
Take confluences of holes to obtain cusps.
Develop bordered cusped Teichmuller theory asymptotically.
This will provide cluster algebra of geometric type
Marta Mazzocco Quantum cluster algebras from geometry
Cluster algebras Teichmuller Theory Geodesic lengths Quatisation Decorated character variety
Poincare uniformsation
Σ = H/∆,
where ∆ is a Fuchsian group, i.e. a discrete sub-group of PSL2(R).
Examples
γ1γ2
Marta Mazzocco Quantum cluster algebras from geometry
Cluster algebras Teichmuller Theory Geodesic lengths Quatisation Decorated character variety
Poincare uniformsation
Σ = H/∆,
where ∆ is a Fuchsian group, i.e. a discrete sub-group of PSL2(R).
Examples
γ1γ2
Theorem
Elements in π1(Σg ,s) are in 1-1 correspondence with conjugacyclasses of closed geodesics.
Marta Mazzocco Quantum cluster algebras from geometry
Cluster algebras Teichmuller Theory Geodesic lengths Quatisation Decorated character variety
Coordinates: geodesic lengths
Theorem
The geodesic length functions form an algebra with multiplication:
GγGγ = Gγγ + Gγγ−1 .
γ
γ
Marta Mazzocco Quantum cluster algebras from geometry
Cluster algebras Teichmuller Theory Geodesic lengths Quatisation Decorated character variety
Coordinates: geodesic lengths
Theorem
The geodesic length functions form an algebra with multiplication:
GγGγ = Gγγ + Gγγ−1 .
γ
γ
=
Marta Mazzocco Quantum cluster algebras from geometry
Cluster algebras Teichmuller Theory Geodesic lengths Quatisation Decorated character variety
Coordinates: geodesic lengths
Theorem
The geodesic length functions form an algebra with multiplication:
GγGγ = Gγγ + Gγγ−1 .
γ
γ
=
γγ−1
+
γγ
Marta Mazzocco Quantum cluster algebras from geometry
Cluster algebras Teichmuller Theory Geodesic lengths Quatisation Decorated character variety
Poisson structure
{Gγ ,Gγ} =1
2Gγγ −
1
2Gγγ−1 .
{γ
γ
}= 1
2
γ−1γ
−12
γγ
Marta Mazzocco Quantum cluster algebras from geometry
Cluster algebras Teichmuller Theory Geodesic lengths Quatisation Decorated character variety
Two types of chewing-gum moves
Connected result:
Disconnected result:
Marta Mazzocco Quantum cluster algebras from geometry
Cluster algebras Teichmuller Theory Geodesic lengths Quatisation Decorated character variety
Chewing gum
z1z2z3
1
1 + ε
εℓ1 εℓ2
(sinh dH(z1,z2)
2
)2= |z1−z2|2
4ℑz1ℑz2
edH(z1,z2) ∼ 1l1l2ϵ2
+ (l1+l2)2
l1l2+O(ϵ),
edH(z1,z3) ∼ edH(z1,z2) + 1l1l2
+O(ϵ).
⇒ Rescale all geodesic lengths by eϵ and take the limit ϵ → 0.[Chekhov-M.M. arXiv:1509.07044]
Marta Mazzocco Quantum cluster algebras from geometry
Cluster algebras Teichmuller Theory Geodesic lengths Quatisation Decorated character variety
γe γf
=skein
γa
γc
+ + the rest
γb γd
γa
γc
γb γdγeγf GγeGγf = GγaGγc + GγbGγd
Marta Mazzocco Quantum cluster algebras from geometry
Cluster algebras Teichmuller Theory Geodesic lengths Quatisation Decorated character variety
γe γf
=skein
γa
γc
+ + the rest
γb γd
γa
γc
γb γdγeγf GγeGγf = GγaGγc + GγbGγd
Marta Mazzocco Quantum cluster algebras from geometry
Cluster algebras Teichmuller Theory Geodesic lengths Quatisation Decorated character variety
Poisson bracket
Introduce cusped laminations
Compute Poisson brackets between arcs in the cuspedlamination.
Theorem
Given a Riemann surface of any genus, any number of holes and atleast one cusp on a boundary, there always exists a completecusped lamination [Chekhov-M.M. ArXiv:1509.07044].
Marta Mazzocco Quantum cluster algebras from geometry
Cluster algebras Teichmuller Theory Geodesic lengths Quatisation Decorated character variety
Poisson structure
Theorem
The Poisson algebra of the λ-lengths of a complete cuspedlamination is a cluster algebra [Chekhov-M.M. ArXiv:1509.07044].
{gsi ,tj , gpr ,ql} = gsi ,tjgpr ,qlIsi ,tj ,pr ,qlIsi ,tj ,pr ,ql =
ϵi−r δs,p+ϵj−r δt,p+ϵi−lδs,q+ϵj−lδt,q4
Marta Mazzocco Quantum cluster algebras from geometry
Cluster algebras Teichmuller Theory Geodesic lengths Quatisation Decorated character variety
Quantisation
For standard geodesic lengths Gγ → G ℏγ [Chekhov-Fock ’99]:
[G ℏγG ℏ
γ
]= q−
12
G ℏγ−1γ
+ q12
G ℏγγ
[G ℏγ ,G
ℏγ ] = q−
12G ℏ
γ−1γ + q12G ℏ
γγ
For arcs gsi ,tj → gℏsi ,tj
:
qIsi ,tj ,pr ,ql gℏsi ,tj
gℏpr ,ql
= gℏpr ,ql
gℏsi ,tj
qIpr ,ql ,si ,tj
This identifies the geometric basis of the quantum clusteralgebras introduced by Berenstein and Zelevinsky.
Marta Mazzocco Quantum cluster algebras from geometry
Cluster algebras Teichmuller Theory Geodesic lengths Quatisation Decorated character variety
Decorated character variety
What is the character variety of a Riemann surface with cusps onits boundary?For Riemann surfaces with holes:
Hom (π1(Σ),PSL2(C)) /GL2(C).
For Riemann surfaces with bordered cusps:Decorated character variety [Chekhov-M.M.-Rubtsov arXiv:1511.03851]
Replace π1(Σ) with the groupoid of all paths γij from cusp ito cusp j modulo homotopy.
Replace tr by two characters: tr and trK .
Marta Mazzocco Quantum cluster algebras from geometry
Cluster algebras Teichmuller Theory Geodesic lengths Quatisation Decorated character variety
Shear coordinates in the Teichmuller space
Fatgraph:
8 LEONID CHEKHOV, MARTA MAZZOCCO, VLADIMIR RUBTSOV
s3
p3s1
p1
s2
p2
Figure 4. The fat graph of the 4 holed Riemann sphere. The reddashed geodesic is x1.
are obtained by decomposing each hyperbolic matrix γ ∈ ∆g,s into a product ofthe so–called right, left and edge matrices:
R :=
!1 1−1 0
", L :=
!0 1−1 1
", Xsi :=
!0 − exp
#si2
$
exp#− si
2
$0
".
In this setting our x1, x2, x3 are the geodesic lengths of thee geodesics which goaround two holes without self–intersections.
We are now going to produce a similar shear coordinate description of each of theother Painleve cubics. We will provide a geometric description of the correspondingRiemann surface and its fat-graph. Our geometric description agrees with the oneobtained in [32], which was obtained by building a Strebel differential from theisomonodromic problems associated to each of the Painleve equations.
3.1. Shear coordinates for PV . The confluence from the cubic associated toPVI to the one associated to PV is realised by
p3 → p3 − 2 log[ϵ],
in the limit ϵ → 0. We obtain the following shear coordinate description for thePV cubic:
x1 = −es2+s3+p22+
p32 −G3e
s2+p22 ,
x2 = −es3+s1+p32+
p12 − es3−s1+
p32−
p12 −G3e
−s1−p12 −G1e
s3+p32 ,
x3 = −es1+s2+p12+
p22 − e−s1−s2−
p12−
p22 − es1−s2+
p12−
p22 −G1e
−s2−p22 −G2e
s1+p12 ,
(3.12)
where
Gi = epi2 + e−
pi2 , i = 1, 2, G3 = e
p32 , G∞ = es1+s2+s3+
p12+
p22+
p32 .
These coordinates satisfy the following cubic relation:
x1x2x3 + x21 + x2
2 − (G1G∞ +G2G3)x1 − (G2G∞ +G1G3)x2 −
−G3G∞x3 +G2∞
+G23 +G1G2G3G∞ = 0.(3.13)
Decompose each hyperbolic element in Right, Left and Edgematrices Fock, Thurston
R :=
(1 1−1 0
), L :=
(0 1−1 −1
),
Xy :=
(0 − exp
( y2
)exp
(− y
2
)0
).
Marta Mazzocco Quantum cluster algebras from geometry
Cluster algebras Teichmuller Theory Geodesic lengths Quatisation Decorated character variety8 LEONID CHEKHOV, MARTA MAZZOCCO, VLADIMIR RUBTSOV
s3
p3s1
p1
s2
p2
Figure 4. The fat graph of the 4 holed Riemann sphere. The reddashed geodesic is x1.
are obtained by decomposing each hyperbolic matrix γ ∈ ∆g,s into a product ofthe so–called right, left and edge matrices:
R :=
!1 1−1 0
", L :=
!0 1−1 1
", Xsi :=
!0 − exp
#si2
$
exp#− si
2
$0
".
In this setting our x1, x2, x3 are the geodesic lengths of thee geodesics which goaround two holes without self–intersections.
We are now going to produce a similar shear coordinate description of each of theother Painleve cubics. We will provide a geometric description of the correspondingRiemann surface and its fat-graph. Our geometric description agrees with the oneobtained in [32], which was obtained by building a Strebel differential from theisomonodromic problems associated to each of the Painleve equations.
3.1. Shear coordinates for PV . The confluence from the cubic associated toPVI to the one associated to PV is realised by
p3 → p3 − 2 log[ϵ],
in the limit ϵ → 0. We obtain the following shear coordinate description for thePV cubic:
x1 = −es2+s3+p22+
p32 −G3e
s2+p22 ,
x2 = −es3+s1+p32+
p12 − es3−s1+
p32−
p12 −G3e
−s1−p12 −G1e
s3+p32 ,
x3 = −es1+s2+p12+
p22 − e−s1−s2−
p12−
p22 − es1−s2+
p12−
p22 −G1e
−s2−p22 −G2e
s1+p12 ,
(3.12)
where
Gi = epi2 + e−
pi2 , i = 1, 2, G3 = e
p32 , G∞ = es1+s2+s3+
p12+
p22+
p32 .
These coordinates satisfy the following cubic relation:
x1x2x3 + x21 + x2
2 − (G1G∞ +G2G3)x1 − (G2G∞ +G1G3)x2 −
−G3G∞x3 +G2∞
+G23 +G1G2G3G∞ = 0.(3.13)
The three geodesic lengths: xi = Tr(γjk)
x1 = es2+s3 +e−s2−s3 +e−s2+s3 +(ep22 +e−
p22 )es3 +(e
p32 +e−
p32 )e−s2
x2 = es3+s1 +e−s3−s1 +e−s3+s1 +(ep32 +e−
p32 )es1 +(e
p12 +e−
p12 )e−s3
x3 = es1+s2 +e−s1−s2 +e−s1+s2 +(ep12 +e−
p12 )es2 +(e
p22 +e−
p22 )e−s1
{x1, x2} = 2x3 + ω3, {x2, x3} = 2x1 + ω1, {x3, x1} = 2x2 + ω2.
Marta Mazzocco Quantum cluster algebras from geometry
Cluster algebras Teichmuller Theory Geodesic lengths Quatisation Decorated character variety
PV
PV flipped:
PVdeg:
γb = X (k1)RX (s3)RX (s2)RX (p2)RX (s2)LX (s3)LX (k1)
BUT its length is b = trK (γb) = tr(bK ), K =
(0 0−1 0
)
Marta Mazzocco Quantum cluster algebras from geometry
Cluster algebras Teichmuller Theory Geodesic lengths Quatisation Decorated character variety
PV
PV flipped:
PVdeg:
{gsi ,tj , gpr ,ql} = gsi ,tjgpr ,qlϵi−r δs,p+ϵj−r δt,p+ϵi−lδs,q+ϵj−lδt,q
4
{b, d} = {g13,14 , g21,18}
= g13,14g21,18ϵ3−1δ1,2 + ϵ4−1δ1,2 + ϵ3−8δ1,1 + ϵ4−8δ1,1
4
= −bd1
2
Marta Mazzocco Quantum cluster algebras from geometry
Cluster algebras Teichmuller Theory Geodesic lengths Quatisation Decorated character variety
Mutations
Example
Riemann sphere with three holes, and two cusps on one of theholes. Frozen variables: c , d , e. Exchangeable variables: a, b.
28 LEONID CHEKHOV, MARTA MAZZOCCO, VLADIMIR RUBTSOV
any other branch of that solution will corresponds to points (y1, y2, y3) on the samecubic such that each yi is a Laurent polynomial of the initial (y01 , y
02 , y
03).
5.2. Generalised cluster algebra structure for PV and PVdeg. In this case,we have a Riemann surface Σ0,3,2 with two bordered cusps on one hole. The onlynontrivial Dehn twist is around the closed geodesic γ encircling these two holes (thisgeodesic is unique). Its geodesic function Gγ is the Hamiltonian MCG invariant.We now consider the effect of this MCG transformation on the system of arcs inFig. 6.
e d
c
b aγ
ω2 ω1
Ma
e d
c
b
a′ω2 ω1
Mb
e d
b′
a′
c
ω2 ω1
Here the generalized mutations Ma and Mb are given by the formulas
a′a = b2 + c2 + ω1bc; b′b = (a′)2 + c2 + ω2a′c,
or, explicitly,
(5.60)
[ab
]→
⎡
⎢⎣
b2 + c2 + ω1bc
a(b2 + c2 + ω1bc)2
a2b+ ω2c
b2 + c2 + ω1bc
ba+
c2
b
⎤
⎥⎦ .
The geodesic function of γ is
(5.61) Gγ = ω2c
b+ ω1
c
a+
a
b+
b
a+
c2
aband this function is the (Hamiltonian) MCG invariant: it generates the corre-sponding Dehn twist (see [19]), has nontrivial Poisson brackets with a and b, andis preserved by the MCG action (5.60).
In the case of PVdeg, all the above formulas remain valid provided we replace cby the λ-length d of the boundary arc.
5.3. Generalised cluster algebra structure for PIIID6 , PIIID7 , and PIIID8 .In all cases of PIII, we have a Riemann surface Σ0,2,n with n1 > 0 and n2 > 0,n1 + n2 = n, bordered cusps on the respective holes. For any n1 and n2, the onlynontrivial Dehn twist is around the closed geodesic γ separating the holes (thisgeodesic is unique). Its geodesic function Gγ is the Hamiltonian MCG invariant.Besides this invariant, we have (non-Hamiltonian) invariants, which are λ-lengthsof all arcs starting and terminating at the same boundary component.
a = g15,16 , b = g13,14 , d = g18,22 , {a, b} = ab, {a, d} = −ad2 .
Sub-algebra of functions that commute with the frozen variablesChekhov-M.M.-Rubtsov arXiv:1511.03851:
{x1, x2} = 2x3 + ω3, {x2, x3} = 2x1 + ω1, {x3, x1} = 2x2 + ω2.Marta Mazzocco Quantum cluster algebras from geometry
Cluster algebras Teichmuller Theory Geodesic lengths Quatisation Decorated character variety
Conclusion
A Riemann surface of genus g , n holes and k cusps on theboundary admits a complete cusped lamination of6g − 6 + 2n + 2k arcs which triangulate it.
Any other cusped lamination is obtained by the cluster algebramutations.
By quantisation: quantum cluster algebra of geometric type.
New notion of decorated character variety
Many thanks for your attention!!!
Marta Mazzocco Quantum cluster algebras from geometry
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